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Gerbault, Muriel At what stress level is the central Indian Ocean lithosphere
buckling? (2000) Earth and Planetary Science Letters, 178 (1-2). ISSN 0012-821X
At what stress level is the central Indian Ocean
lithosphere buckling?
Muriel Gerbault
Institute of Geological and Nuclear Sciences, P. O. Box 30 368 Lower Hutt, New Zealand
Abstract
Interna] contrasts in strength are responsible for lithospheric buclding. These are quantified by comparing the Indian Oœan data to two dimensional visco elasto plastic numerical models where the material properties depend on temperature and pressure. The central Indian Basin is known for its intraplate seismicity and long wavelength undulations of the sea floor and associated gravity signatures. To simula te the amplitudes of undulations that reach 1 km within about 11 Ma of compression and 60 km of shortening, the required mean yield strength of the lithosphere is 400 MPa. If either a hydrostatic fluid pressure is considered in the crust or a mechanical decoupling at the depth of the Moho, small crustal wavelengths are superimposed on the long wavelength deformation, in agreement with observations. It is then possible to match the alternative indication that buckling co=enced only 4 Ma ago, with a total amount of shortening of 30 km, and with a required yield strength of 200 MPa. About 10% of homogeneous thickening accompanies buckling. Taking into account variable thermal conductivity demonstrates that a change in the geotherm is sufficient to increase buckling amplitudes by 10% within 5 Ma. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: lndian Oœan; intraplate processes; stress; plastic deformation; geodynam ics; rheology; serpentinite
1. Introduction
The Cretaceous oceanic lithosphere in the cen tral lndian Basin has an unusual amount of intra plate seismicity. Free-air gravity anomalies and sea floor topography undulate with a wavelength of about 200 km, over distances exceeding 1000 km [1]. These features are an expression of litho spheric scale buckling, which has been well de scribed in a number of analytical [2,6], analog [7,8], and numerical models [9, 10].
Although the term 'buckling' is classically re Jated to an elastic mode of deformation, it may also be used to describe deformation under a
plas-tic (i.e. irreversible) yield stress, with a similar meaning to plastic folding. Resolution of the dif ferential equation of deflection of a plate yields stresses that are too high if one assumes that the lithospheric rheology is elastic [15,16]. Whilst Newtonian viscous rheology provides time scales that are too short [17], a depth dependent power Jaw rheology appears to be more appropriate [18, 19]. Buckling may develop in a system of stratified Jayers when the contrasts in strength, or effective viscosity, are greater than 103. This
mechanism involves Jess energy than if deforma tion remains homogeneous (Fig. l a). Analytical solutions are scale independent and are
character-ised by the growth rate associated with a wave-length V: the most likely wavelength to develop depends on the thickness h such that V/h = 4 6 (Fig. 1b) [2,3,5,18 20].
The e¡ect of gravity increases with the thick-ness of the system; it a¡ects the growth rate of buckling instabilities, and thus the geometry of deformation. Using the assumption of visco-elas-ticity, Lambeck [6] predicted that faults can ini-tiate at the in£ection of the folds, because of the dependency of the vertical stress on the vertical de£ection. Analog modellings, which can handle large deformation, reproduce such a geometry [7,8] with buckling appearing after 1% of short-ening, and faults stabilising at the in£ection points after 5% of shortening [21]. It was also shown numerically that the lithosphere undergoes di¡use faulting prior to buckling [10,22]. Subsequently, faults and folds develop together in coherent kine-matics (Fig. 1c).
The more initial perturbations are present, the easier di¡erent wavelengths may grow and lead to an irregular geometry of folds [23]. Thus buckling in the Indian Ocean is a rare feature among litho-spheres, since it requires: (i) a uniform and par-ticular high distribution of stresses, and (ii) rela-tively few initial perturbations. It is an ideal site for studying the maximum shear stress that can be supported by an intraplate oceanic lithosphere.
In this contribution, wavelength, amplitude and timing of buckling in the Indian Ocean sphere are used as a stress-meter of the litho-sphere and compared with forward numerical cal-culations. A yield strength envelope (deviatoric stress as a function of depth) can be de¢ned using constitutive laws extrapolated from laboratory ex-periments on rocks [11,12]. Both the (pressure pendent) Mohr Coulomb and (temperature de-pendent) power-law creep constitutive laws give an upper bound for the deviatoric stresses, respec-tively along the average top 15 km of the brittle lithosphere, and below, within the ductile litho-sphere. Other constitutive laws such as di¡usion creep or Peierls mechanism [11 14] could approx-imate better the deformation that occurs at depths around the brittle ductile transition (BDT). How-ever, this paper only deals with the two ¢rst laws because it concentrates on the stress ¢eld aspect.
The ¢rst section presents the data and uncer-tainties from the central Indian Basin, which are used to compare with six di¡erent models. The ¢rst models (1, 2 and 3) demonstrate the role of elasto brittle parameters on the growth rate of buckling. The remaining models 4 and 5 take into account the presence of £uids in the crust. These allow for independent buckling of the crust and match the observation of 5 20 km of short-wavelength deformation of the Indian Ocean sur-face. The distribution of temperature is crucial in determining the stress ¢eld. Recent work suggests that thermal conductivity may vary signi¢cantly across the lithosphere [24]. One-dimensional mod-eling predicts that variable conductivity can pro-duce changes in surface topography over a few hundred meters [25]. This particular e¡ect is ex-amined in a ¢nal model (6).
2. Deformation and stresses inferred from data 2.1. Quantifying the brittle and folding
deformation
The central Indian Basin was identi¢ed as a di¡use plate boundary between India and Austra-lia [26,27]. The AustraAustra-lian plate has been rotating counterclockwise relative to the Indian plate about a pole located west of the central Indian Basin, east of the Chagos Lacadive ridge [26,27]; approximately north-south convergence increases with increasing distance eastwards from the pole of rotation (Fig. 2A).
Pole rotation reconstructions [28] imply motion at 6 mm a 1 across the central Indian Basin from 20 to 18 Ma. The rate of motion was slow from 18 to 11 Ma (1 mm a 1), increased to 4 mm a 1 between 11 Ma and 3 Ma, and further increased to 6 mm a 1 since 3 Ma [28]. These reconstruc-tions estimate a north-south component of short-ening of 31 þ 7 km along 80³E, and 80 þ 12 km along 90³E [28]. If the di¡use boundary is 900 km (as indicated by seismic re£ection pro¢ling along 81.5³E [29]), the average north-south longitudinal strain is 3.4 þ 0.8%. If it is 2000 km wide at 90³E, the strain is 4 þ 0.4%. The north-south component of convergence across the central Indian Basin
has been estimated independently from the throws and dips of thrust faults observed on seismic pro-¢les [29,30]: Jestin [30] estimated a total shorten-ing of 44 þ 10 km along 84.5³E on a 1000 km seismic line, while pole reconstructions infer 53 þ 6 km. This discrepancy may be due to the fact that small motions are di¤cult to measure in seismic pro¢les.
Heat £ux anomalies reach an average 30 mW m 2 above the theoretical value for an oceanic crust of that age (55 mW m 2) [31]. The authors
suggest that a shortening of 4.5% (50 km for 1100 km) would be enough to produce a heat £ux anomaly of 20 mW m 2. Values exceed 200 mW m 2 in some places [32], and were suggested to reveal the exothermic reaction of serpentinisation due to lithospheric fracturing and water penetra-tion [32]. Blocks of intraplate deformapenetra-tion display anomalous P-wave velocities in the lower crust, which may indicate the presence of serpentinised peridotites [33].
Earthquakes east of 86³E between 10³S and Fig. 1. (a) Ways to accommodate shortening: by a component of homogeneous thickening and a component of buckling. (b) An alytical growth rate of buckling instabilities as a function of wavelength to thickness ratio, for a given contrast in power law vis cosity (Burov, personal communication). (c) Buckling mode of deformation for an oceanic lithosphere [21]. A preliminary stage of homogeneous thickening is followed by the onset of buckling once the competent layer reaches the yield stress. Accompanying the development of folds, faults simultaneously cut the whole layer at the in£exion points, until one overtakes on the others and initiates subduction.
10³N typically have strike-slip mechanisms with a few small events with reverse mechanisms, where-as earthquakes west of 86³E in the southern Bay of Bengal have thrust faulting mechanisms (a thrusting focal mechanism was recorded at 39 km depth) [34]. Faults observed along seismic pro¢les are associated with re£ectors that cross the Moho horizon [21,31], with no preferred dip-ping direction nor steepness. Some, but not all, crustal thrust faults are interpreted as pre-existing spreading ridge normal faults [29,30].
Series of tight folds and high angle faults of a
spacing of 5 20 km are widely distributed and are superposed to the long-wavelength folds of 150 300 km, which trend perpendicular to the direc-tion of compression [21,35] (Fig. 2B,C). This pe-riodicity extends along more than 1000 km. Jestin [30] argued for an anti-correlation between rises of the Moho horizon and rises of the top of the crust, suggesting inverse boudinage instead of buckling of the crust. Her discussion was based on the concentration of maximum vertical defor-mation at the crest of some of the folds (Fig. 2D). Le¨ger and Louden [36] supported the idea that the Fig. 2. (A) Location of the chron 5 Euler poles. The two independent strippled areas show the approximate extent of deforma tion that constitute the boundary between the Australian and Indian plates [29]. (B) Topography and free air gravity from [35]. (C) Approximate north south seismic pro¢le from [35]. (D) Pro¢les of deformation, topography and Moho horizon along 84.5³E [30].
crust is thickened beneath some anticlinal rises and may undergo boudinage. But their gravity and seismic analysis was not supported by Ne-prochnov et al.'s [33] analysis. This controversy was subsequently discussed by [21], and shows that there are yet no data allowing to relate ex-plicitly the geometry of faulting, the location of earthquakes and that of the short- and long-wave-length folds.
The estimate of the stress level at which the Indian lithosphere is buckling requires to know when compression started, thus the time necessary for stresses to build up prior to buckling (further referred to as t1), and the time spanned since buckling develops. As cited above, the rate of motion is believed to have become signi¢cant at about 11 Ma [28]. Three nonconformities are rec-ognised from deep-sea drilling and seismic strat-igraphy across the central Indian Basin [35,37]: a major one at 7.5 8 Ma was suggested to mark the onset of deformation, while the two others are secondary, at 3 4 and 0.8 Ma. The deformation activity could be episodic with a cyclecity of V3.5 Ma, due to a periodic release of stresses within the Indian plate [35]. The 7.5 8 Ma nonconformity, which has been drilled in places separated by about 1000 km (DSDP site 218 and ODP Leg sites 116), may coincide with the onset of normal faulting in the Tibetan plateau, and the time when it attained its maximum elevation. Moreover, dis-turbed oceanic basement and undeformed overly-ing sediments show that there may have been de-formation activity before 8 Ma (early Miocene) [35], at least south of 1³S. Buckling is believed to have started 8 Ma ago south of 1³S [35]. The 3.5 4 Ma nonconformity would mark the onset of buckling north of 1³S [35]. The forces acting be-fore 8 Ma might have been too small or in the wrong direction to cause large scale folding. 2.2. Estimating stresses
Several places within oceanic plates are thought to be in a state of buckling because of elevated compressive stresses, that have built up because of a complex geometry of subductions at the plate boundaries (Philippine plate and the Zenisu ridge, S. Lallemand, personal communication, [38]).
Subduction may occur along these boundaries with di¡erent speeds, so that a transform fault accommodates the horizontal di¡erential move-ments. One area is kinematically blocked and ac-cumulates stresses (Fig. 3). The displacement of India towards north is resisted by the India Asia collision, whereas the eastern part of the Indo Australian plate subducts freely under the Java Sumatra trench. The preexisting Ninetyeast ridge may now accommodate this di¡erential ve-locity.
Models using plane stresses, focal mechanisms, and forces applied at the plate's boundaries pro-vide a ¢rst approximation of the state of stress in the lithosphere. Cloetingh and Wortel [39] pre-dicted a mean compressive stress within the Indo Australian plate, for a plate 100 km thick, a Young's modulus E = 7.1010Pa, and a Poisson's ratioX = 0.25. They ¢nd an average stress of 300 600 MPa in the Indian Ocean Basin. A recent model, based on the same method but with var-iations in the forces acting on the Indo Austral-ian plate boundaries, proposes an average stress of 100 MPa [40].
The intraplate tectonic force can be estimated for an oceanic lithosphere that undergoes buck-ling, because we know that it must reach its fail-ure stresses prior to the development of buckling. Because the age of the central Indian Ocean litho-sphere ranges from 58 to 84 Ma [41], the thickness of the competent lithosphere can be deduced from the heat conduction equation [42]: hV40 km. h can also be associated to the depth at which stresses still sustain more than 5% of the litho-static stress, and can be assimilated to the BDT, which is in agreement with the distribution of seismicity at depths 30 þ 10 km. Nevertheless
Fig. 3. Tectonic setting for generation of buckling in an ocean lithosphere (see text).
earthquakes can occur at a shallower depth than the BDT [43].
The pattern of loading is elastic within the com-petent lithosphere, until the failure stresses are reached. Suppose that the minimum compressive stress c1 is negative, horizontal, and increases down to the BDT, while c3=bgh (b = 3200 kg m 3 is density and g = 10 m2s 1 is gravity accel-eration) is the vertical lithostatic stress (Fig. 4). The Mohr Coulomb relation for no cohesion, d = tan Pcn (P the friction angle, d and cn the shear and normal stresses acting along the failure plane), can be rewritten in terms of principal stresses:
c11 sinP1 sinP c3 1
With the last part of elastic lithosphere lying at depth h = 40 km, and then assuming a common value of P = 30³, one obtains a maximum value c1= 3bgh = 3.8 GPa. Hooke's law for plain strain provides two equations (with Lame¨ parametersVL and G):
c1 VL 2GO1VLO3;
c3 VL 2GO3VLO1 2
Lame¨'s parameters have identical values if Pois-son's ratio X = 0.25. By inserting Eq. 2 into Eq. 1 yields a formula of O3 as a function of O1. By
inserting the expression of O3 back into the ¢rst equation of Eq. 2, one obtains c1= 3WGO1= 3 GW _O1T1.
If 1000 km of plate are compressed at 6 mm a 1 (50 km of shortening within 8 Ma), then the rate at which shortening occurs is _OxxV _O1= 2U10 16 s 1 (this shortening rate is further referred to as _O1). For a modulus of ri-gidity G = 6U1010 Pa (appropriate for oceanic lithosphere), the required time of loading is T1= 3.4 Ma. If either the shortening rate or G are smaller, then T1 increases. If a threshold of the yield stress exists at depth such that c1 is smaller, then T1 is reduced. This value can be compared to the 3 Ma spanned in between 11 and 8 Ma ago [35].
The lateral force required to activate buckling is obtained by integrating the deviatoric stress along depth h = 40 km, for P = 30³: F Z 40 km 0 c1 c3 2 dy Z 40 km 0 1 sinP 1 sinP 1 bgy 2 dy 2:56U1013N m 1
If one considers an alternative stress envelope such that friction is null and cohesion So= 200 MPa along a depth of 30 km, then the average compressive stress is smaller and F = 6U1012 N m 1. Estimates of forces involved in plate tecton-ics only have an order of magnitude of precision. The horizontal di¡erential force required to pre-vent the collapse of high plateaus such as the Tibet plateau [15] ranges from 4U1012 N m 1 to 1013 N m 1. Furthermore, Fowler [44] esti-mates the slab pull to be of the order of 1013 N m 1. The undulations of the geo|«d at trenches indicate a similar value [45].
The Bengal Fan sediments in the northern part of the central Indian Basin facilitate the develop-ment of buckling: because of an intermediate den-sity at the upper interface of the lithosphere, sedi-ments diminish the inhibiting e¡ect of gravity forces on the growth rate of buckling instabilities [2,4]. This explains the higher amplitudes and wavelength in the north with respect to the south (amplitudes reach 2 km and wavelengths reach Fig. 4. Mohr circle and approximate distribution of principal
stresses according to the yield stress envelope for lateral com pression, where bgy is the vertical lithostatic stress (density b, acceleration g, depth y 6 0). The plain line is the stress en velope for non zero cohesion So and friction P. The dashed
300 km), which are also due to the increasing age of the lithosphere to more than 80 Ma. Martinod and Molnar [4] used a perturbation method to estimate that a force per unit length as small as 3.5U1012 N m 1 was enough to fold the Indian Ocean lithosphere, when a density contrast of 1000 kg m 3 due to sediments was included on the upper interface. Without sediments, a di¡er-ential force of around 1013N m 1 is needed. This study implies that the mean strength of the litho-sphere lies in the range 90 350 MPa[4]. Although the most prominent folds are in the Bengal Fan area, periodic structures are also imaged in places of very few sediment deposits south of 1³S, and are even suggested to have developed earlier than those to the north of 1³S [35]. Therefore we need to estimate the distribution of stresses without taking into account the presence of sediments.
Can the following numerical modeling support an onset of deformation 8 or 11 Ma ago? The aim of the following models is to ¢t a realistic rheol-ogy for the lithosphere and provide some clues to answer this question. For this, the geophysical setting is simpli¢ed. We consider a homogeneous age of the lithosphere of 60 Ma along 1000 km, and aim to produce buckling instabilities of 150 200 km wavelength and 1 km amplitude. Results will be compared with the total amount of short-ening estimated in the area, thus 30 km (along 80³E) to 70 km (along 90³E). By assuming that the rate of compression is 6 mm a 1, two main scenarios are considered:
Scenario A refers to 11 Ma of compression (70
km of shortening) and to the development of buckling for 8 Ma. This could validate observa-tions south of 1³S and east of 84.5³E.
Scenario B refers to 30 km of shortening (6 Ma) and to the development of buckling for 4 Ma. This could validate observations north of 1³S and west of 84.5³E.
3. Modeling
3.1. Initial conditions
The numerical program Parovoz written by A.N.B. Poliakov and Y.Y. Podlachikov [46] was used. It is based on the F.L.A.C. method, which incorporates ¢nite di¡erences and an explicit time marching scheme that allows for a wide range of constitutive laws to be included [47].
The lithosphere is modeled as a two-dimension-al medium of 400 by 30 quadrilatertwo-dimension-al elements, with a total length of 1000 km and a depth of 60 km. Constant horizontal velocity is applied on each side of the model, so that _O1= 2U10 16 s 1. All simulations have been run up to about 10% of the total amount of shortening. At the bottom of the model, hydrostatic boundary con-ditions are applied with a density of 3300 kg m 3, while the temperature is prescribed to remain at 1500³C. The surface is stress free. An isotropic pressure of 45 MPa is included to simulate 4.5 km of overlying water.
The temperature is evaluated by using the heat Table 1
Temperature pro¢le from [15], power law creep e¡ective viscosity from [48], parameters from [11]
Temperature Tm Hm U Age (³C) (km) (W m 1K 1) (Ma) T t; y Tmerf y 2 k agep ! 1500 60 3, 3.5, 4.4 60
Creep law (O = 2U10 16 s 1) density n Activation energy A
W 1 4 4 3A 1 n_O 1 n1expH nRT (kg m 3) (J mol 1) (Pa ns 1)
olivine 3200 3. 5.2U105 7U104
plagioclase 2900 3.2 2.38U105 3.3U10 4
Elasto plasticity VL,G (Pa) dhd 6 12 km dhd s 12 km,So dhd s 12 km,P
d = SoWtanPcn 3 6U1010 20 MPap30³ (20; 400 MPa) 0330³
conduction solution [42] for an age of 60 Ma (Table 1). The resulting geotherm leads to a tem-perature of approximately 750³C at the depth of 40 km, and a surface heat £ux of 86 mW m 1 K 1. The temperature is updated with time and shear heating is taken into account. With the ve-locity applied along the sides of the model, an e¡ective viscosity is then obtained from the power-law creep constitutive law (Table 1). Elas-to visco brittle behaviour is modeled such that at each time-step and for each element, the mini-mum deviatoric stress produced by Mohr Cou-lomb elasto-plasticity and visco-elasticity is re-tained.
The in£uence of the pressure of interstitial £u-ids upon the e¡ective stresses in rocks is consid-ered in some of the following models. This is done by modifying the Mohr Coulomb criterion of failure to the form d = tan P(cn bwgh), where bwgh is the hydrostatic pressure at depth h (bw is the water density).
3.2. Models 1, 2, 3: test of elasto brittle parameters
The following models aim to demonstrate the e¡ect of varying the contrast in strength between the competent lithosphere and the lower ductile lithosphere.
3.2.1. Model 1: friction and G = 3U1010 Pa Compressive stresses build up during the ¢rst 7 Ma: failure initiates at the surface and
propa-gates downward (Fig. 5). Di¡use shear bands (they change place from time to time, red on Fig. 6) are distributed through the whole compe-tent layer. Their spacing depends on the thickness of the failed layer, while their thickness depends on the mesh resolution. After 7 Ma, the buckling instabilities start to produce signi¢cant periodic topography (greater than 50 m, Figs. 6 and 7), with a wavelength of 150 200 km. This is in agreement with the rule of thumb for plastic buckling V = 4 6 h, and infers a competent thick-ness of approximately 35 km. As folds develop, shear bands begin to localise at the in£ection points of the folds, and reach the upper interface at the trough of the folds. After 13 Ma, most of the brittle deformation concentrates in these areas. Buckling has been going on for nearly 7 Ma and amplitudes reach only 682 m (Table 2). The associated amount of shortening shows that the growth of buckling instabilities in this model is too slow compared to observations.
Ideally, once the yield strength is reached, the lateral force would be released. But because of the condition of incompressibility, homogeneous thickening leads to thickening of the competent lithosphere and an increase of the integrated de-viatoric stresses. The dede-viatoric force is F = 1.5 1013 N m 1 at the onset of buckling, and reaches a value of 2U1013 after 20 Ma. The competent lithosphere has thickened by approximately 5 km (Fig. 5). This thickening is a consequence of kine-matics and thermal conduction, which processes have not each been quanti¢ed in detail. Because
Fig. 5. Model 1. Temperature, e¡ective viscosity and deviatoric stress ddd 1=4 cxx3cyy2c2xy
q
, at di¡erent moments of the simulation along the section x = 0. The evolution of temperature and viscosity through time are identical for models 2 and 3. No tice the thickening of the competent layer from approximately 35 to 40 km.
of the density contrasts on the top and bottom interfaces, this thickening yields also an increase in the mean elevation of the surface by 200 m (Fig. 7).
3.2.2. Model 2: friction and G = 6U1010 Pa If the modulus of rigidity is 6U1010Pa, which is a value appropriate for dunites [15], then buckling develops sooner than in model 1: amplitudes of 53 m appear after 6 Ma (Table 2 and Fig. 7). After 12.9 Ma, amplitudes reach 922 m, and the amount of shortening is 77 km (Table 2). This
value still exceeds the one obtained from pole rotation reconstructions. In the following models, this value of G is used.
3.2.3. Model 3: So= 400 MPa
Several studies from rock mechanics suggest that the brittle yield stress for depths greater than 15 km becomes independent from pressure [11,12]. When modeling buckling, the main e¡ect of assigning friction and cohesion will be that the less friction, the less faulting localises at the in-£ection points of the folds, because of non-asso-Fig. 7. Evolution of surface topography for several models: maximum, minimum and mean (thick grey lines) amplitudes as a function of time. Left, models 1 and 2 with friction angleP = 30³ for di¡erent values of the modulus of rigidity. Right, models 3,4a,4b,5a,5b considering: a mean yield stress of 400 MPa below 12 km (3), pore £uid pressure above 12 km and 100 200 MPa below 12 km (4a,4b), a low viscosity layer at the crust mantle boundary and 100 200 MPa below 12 km (5a,5b). Because of the density contrasts at the top and bottom interfaces of the model lithosphere, thickening leads to an increase in the mean eleva tion.
6
Fig. 6. Model 1. Second invariant of the strain rate and associated topography at three moments of the simulation: after 7.3 Ma, 12.6 Ma and 16.7 Ma. Di¡use shear bands in red (faults) are distributed across the competent layer. After 16.7 Ma they concentrate in the in£ection point of the folds. Deformation remains di¡use in the lower ductile lithosphere below approximately 35 km.
6
Fig. 9. Model 5a. Low viscosity layer in between 8 and 12 km depth, and So= 200 MPa below 12 km. Topography and second
invariant of the strain rate. High shear strain (red) concentrates in the low viscosity layer. Low values (blue) are located in areas of least compression. Horizontal distances are in km. Because the upper crust is 8 km thick, the crustal wavelength is about 40 km. The long wavelength associated to mantle lithosphere buckling is about 150 km. Right: deviatoric shear stress along the cen tral section of the plate (S = 0).
6
Fig. 8. Model 4a: Pore £uid pressure in the crust and So= 200 MPa below 12 km. Topography and second invariant of the
strain rate after 7.3 Ma. Notice the small undulations on the surface, associated to crustal faulting. Right: deviatoric shear stress along the central section of the plate.
ciated plasticity [49] (localised shear zones depend on the di¡erence in the dilatance and friction an-gles). Model 3 is de¢ned with Mohr Coulomb parameters such that from 0 to 12 km depth, P = 30³, So= 20 MPa. Below 12 km, P = 0³, So= 400 MPa, with other parameters identical to model 2.
Two opposite e¡ects appear: since the yield strength is smaller than in the previous models, buckling initiates sooner. But because the contrast in strength is smaller, buckling growth is slower.
After approximately 2 Ma, the stresses attain their maximum value, and large wavelengths start to develop. Amplitudes reach 1050 m after 10.5 Ma and 62 km of shortening (Table 2). This mod-els could support scenario A. Because of associ-ated plasticity at depth greater than 12 km, no `thin' shear bands localise at the in£ection points of the folds, as there were in model 1.
There are not many ways to accelerate buckling growth. One may increase the contrast in strength by reducing the viscosity of the lower lithosphere. For this, the temperature requires to increase. But then, for the creep parameters of olivine displayed in Table 1, the BDT will be less than what is assumed for the Indian Ocean, and the surface heat £ux will be higher. Increasing the rate of compression is not a solution either since the pre-vious models already require too much shorten-ing.
3.3. Models 4, 5: the role of £uids
The upper part of the lithosphere may be hy-drated, as indicated by the heat £ux anomaly [31,32]. According to [32], `the anomalous rate of heat £ow may be explained by dissipative heat-ing of the overlyheat-ing crustal layer in the process of sliding of the mantle part of the lithosphere below the serpentinite layer in the direction of the Indi-an Asia subduction zone'. Serpentinised olivine might constitute a thin layer of low viscosity and a mechanical decoupling between the crust and mantle. Data report a regular spacing of faults every 5 20 km [21,30,35] that may be re-lated to crustal buckling. Furthermore, bihar-monic folding can occur when two competent layers (such as the crust and the mantle) are sep-arated by a relatively weak layer (i.e. ductile lower crust or serpentinite), and produce independent wavelengths that appear on the surface. This was analytically and numerically demonstrated for continental lithosphere [3,22].
The presence of £uids in the oceanic crust is taken into account in two ways. A ¢rst way is to consider the e¡ective stresses in the Mohr Coulomb criterion of failure, as discussed in Sec-tion 3.1. This is done in models 4. The other way is to simulate the concentration of £uids in a layer located at the crust mantle boundary, so that this layer has a low viscosity. Models 5 take into
ac-Table 2
Timing, shortening, and maximum amplitudes for all models
Model special So Low vis. Time Shortening Amplitude Scenario
(MPa) (Ma) (km) (m) 1 G = 30 GPa,P = 30³ 20 no 12.6 76 682 2 G = 60 GPa,P = 30³ 20 no 12.9 78 922 3 400 no 10.5 62 1050 A 4a pf 200 no 7.3 44 1010 B 4b pf 100 no 11.0 66 780 A 5a 200 yes 5.2 31 990 B 5b 100 yes 6.2 38 1070 5c O1= 10 16 s 1 200 yes 6.3 22 1080 B 6 VariableU 200 yes 5.2 31 1140 B
Except for special cases, G = 60 GPa, P = 0³, and O1= 2U10 16 s 1. Cohesion So is given for the lithospheric mantle below 12
km. The plagioclase rheology is used for models 4, 5 and 6 above 12 km. pf stands for consideration of pore £uid pressure, and
count a low viscosity layer at a depth between 8 and 12 km withW = 2U1020Pa s. It is not thinner because it would require a ¢ner mesh resolution.
Below 12 km, the olivine rheology is still con-sidered, the friction angle is null and the cohesion varies for di¡erent models. Above 12 km, the power law creep parameters are those of a plagio-clase (Table 1), and the yield parameters are So= 20 MPa and P = 30³ (except in the range 8 12 km for models 5). The density is set to 2900 kg m 3 between 0 and 12 km, simulating the lighter gabbro part of the lithosphere. A positive density contrast triggers the development of buckling, be-cause it reduces the action of gravity [2,4,5]. The thermal conductivity varies now with depth: U = 4.4, 3.5, and 3 W m 1 K 1 respectively for the upper crust, the weak viscous layer and the mantle lithosphere.
3.3.1. Models 4: pore-£uid pressure in the crust 3.3.1.1. Model 4a
So= 200 MPa below 12 km. Although the
stresses reach their maximum yield value after ap-proximately 0.6 Ma, large undulations become signi¢cant after 2 Ma (Table 2). Amplitudes reach 1010 m after 7.3 Ma and 44 km of shortening (Fig. 8). This is an ideal model to support scenar-io B. Lithospheric scale buckling develops with a wavelength VV200 km. Small folds have devel-oped with a spacing approximately equal to 10 km, which are still detectable in the troughs of the large-scale folds at the surface (Fig. 8). These are related to faulting restricted to the crustal layer, with an amplitude less than 50 m (this num-ber is an approximation because the mesh size is too big to resolve such small distances). The de-viatoric stress is 40 MPa at 12 km depth, just above the sharp increase to 200 MPa imposed by the cohesion. Free-air gravity anomalies corre-late with the topography, with amplitudes of about 50 mGal after 7.3 Ma.
3.3.1.2. Model 4b
So= 100 MPa below 12 km. With amplitudes that reach 780 m after 11 Ma and more than 66 km of shortening, scenario A is hard to satisfy with this model.
3.3.2. Models 5: low viscosity layer 3.3.2.1. Model 5a
So= 200 MPa below 12 km. A short wavelength instability develops within the ¢rst million year, related to buckling of the crust. The long wave-length related to lithospheric buckling is now re-duced to 150 km. Amplitudes reach 990 m after 31 km of shortening and 5.2 Ma of compression. They reach 1270 m after 38 km of shortening and 6.3 Ma of compression. This model supports sce-nario B. Fig. 9 shows the biharmonic deforma-tion: amplitudes related to the short buckling wavelength are higher than those produced in model 4a. Shear strain concentrates in the low viscosity layer between 8 and 12 km. Because the upper crust is chosen to be 8 km thick, the associated wavelength is about 40 km (5 h). 3.3.2.2. Model 5b
So= 100 MPa below 12 km. Amplitudes on the surface are again close to 1 km after 6.2 Ma of shortening, but they are related to the crustal buckling only. The long wavelength does not de-velop. This shows that the contrast in strength across the lithospheric mantle is too small and that it rather deforms homogeneously, while the crust buckles intensively (stresses reach a maxi-mum yield stress of 200 MPa at the depth of 8 km).
3.3.2.3. Model 5c
So= 200 MPa below 12 km and the rate of compression is divided by 2 ( _O1= 10 16 s 1). The time interval for elastic stresses to build up within the competent layer is twice as long as it is in model 5a. Amplitudes reach 1080 m after 22 km of shortening and 6.3 Ma of compression (only 200 m less than in model 5a after a similar time).
By introducing a low viscosity layer, the long wavelength related to lithospheric buckling is re-duced, because unlikely to models 4, the top 12 km are eliminated from the thickness of the layer that is involved in lithospheric buckling (see the stress envelopes on Fig. 8 and 9). Because each process (pore-£uid pressure and low viscosity layer) leads to the development of di¡erent
large-scale wavelengths (200 and 150 km, respec-tively), their combined e¡ect slows down the de-velopment of a single one. Therefore only one of those two ways to take £uids into account should be considered, and not both.
3.3.3. Model 6: variable thermal conductivity Recent work [24] shows that the thermal con-ductivity can be estimated from the core to the surface of the Earth, by considering the infrared re£ectivity of phonons lifetimes. For lithospheric depths, the variable thermal conductivity takes the form:
k P; T k298 298T
a
exp 1 K 4 QT13T 1 QPP Where k298= 4.4 W m 1 K 1 is the thermal con-ductivity at 298 K, a = 0.33 is a power-law expo-nent, K = 2U10 5 K 1 is the thermal expansivity,
QT= 1.4 is a Gruneisen parameter, QP=
3.51U10 11 Pa 1 is the ratio of isothermal bulk modulus and its pressure derivative, P is the litho-static pressure and T is the temperature (values are given according to the olivine rheology at am-bient conditions). In Parovoz, the temperature is resolved and updated during timesteps within each element. It is modi¢ed by adding the two last terms of the temperature equation:
DT
Dt k ! 92T DTDk 9T2DPDk9P 9T
For the present setting, the calculated initial thermal conductivity varies from 4.4 W m 1 K 1 at the surface to 2.2 W m 1 K 1 at 60 km depth. There is a feedback between temperature and thermal conductivity, such that they both evolve through time. Increase in pressure within the brittle layer has an e¡ect as well. Results are presented in Table 2, and show that in com-parison with model 5a, the amplitude is ampli¢ed by a few hundred meters within 5 Ma of short-ening. A variation of thermal conductivity with depth modi¢es enough the yield envelope so as to produce similar di¡erences of the growth rate of buckling instabilities. This example
illus-trates the sensitivity of the results to initial con-ditions.
4. Conclusions
The wavelength, amplitudes and timing of de-formation in the Indian Ocean Basin are com-pared to numerical simulations of lithospheric-scale buckling. This allows one to place reason-able bounds on the short term (10 Ma time frame) rheological properties of the oceanic lithosphere. The upper brittle layer is modelled by Coulomb plasticity and the lower ductile layer by power-law creep. The contrast in strength between competent brittle lithosphere and deeper ductile lithosphere can be determined as it is bound by two e¡ects. If the contrast is set too high in the models, the amplitudes grow faster than is observed. If the contrast is set too low, then the amount of short-ening is greater than observations. Additional models simulating the presence of a hydrated crustal layer use an oceanic strength pro¢le that is similar to the continental Christmas-tree stress envelope. Then, deformation displays a second shorter wavelength that coexists with the litho-spheric-scale wavelength.
The geophysical measurements used to con-strain the numerical models were simpli¢ed into two extreme scenarios. To produce long-wave-length amplitudes of 1 km, either more than 60 km of shortening and 11 Ma of compression are required, which is referred to as scenario A, or 30 km of shortening and 6 Ma of compression are required, which is referred to as scenario B. This study shows that:
1. Prior to the development of lithospheric buck-ling, stresses exhibit an elastic behaviour and `build up' for a time t1=c1/3G _O1. Taking c1 related to the mean deviatoric shear stress d = 400 MPa, a modulus of rigidity G = 6U1010 Pa, and a shortening rate _O1= 2U10 16 s 1 (6 mm a 1 along 1000 km), gives t1= 2.6 Ma. This time interval coincides with geophysical data that suggest an onset of com-pression 11 Ma ago in the central Indian Ba-sin, and the development of buckling folds
8 Ma ago, south of 1³S [35]. Model 3, with a mean yield stress of 400 MPa, veri¢es scenario A and this time interval t1. The associated mean stress value is consistent with the results obtained by Cloetingh and Wortel [39] (300 600 MPa).
2. The presence of £uids was taken into account ¢rst by including the pore-£uid pressure in the Mohr Coulomb failure criterion (models 4), and then by supposing that a layer of low vis-cosity decouples the crust mantle boundary (models 5). Scenario A is satis¢ed for a litho-spheric mantle strength of 200 MPa (models 4a and 5a). Scenario B is satis¢ed with di¤culty if the mean yield stress is as small as 100 MPa (model 4b), because the compression is rather accommodated homogeneously. These models give stress values closer to the results from Co-blentz et al. [40]. In their study, the horizontal force is averaged over a 100 km thick elastic plate. To scale it to our 40 km thick competent lithosphere, their mean stress values need to be doubled, giving 200 MPa for the central Indian Basin. A thin serpentinite layer at the crust mantle boundary, as was included in models 5, ¢ts well with geophysical measurements. However its existence remains unproven, apart from anomalous P-wave velocities reported in the lower crust [33]. A decoupled sliding of the mantle lithosphere towards the Indo Asia sub-duction zone might be responsible for the propagation into the oceanic lithosphere, of a similar decollement at the crust mantle bound-ary [32]. But £exural studies show that volca-noes are supported by much thicker elastic oce-anic lithosphere (generally about 30 km) [15]. If a weak layer exists within the top 10 km, either it is not thick enough to a¡ect the large scale elastic properties of the oceanic litho-sphere, or it is not a general property of all oceanic lithospheres. Therefore models 4 with a pore-£uid pressure might be the most reason-able way to simulate the e¡ect of £uids in the oceanic crust.
3. Martinod and Molnar [4] demonstrated that the presence of a low-density layer of sedi-ments can reduce the mean yield strength to 90 MPa, su¤ciently low to produce buckling
instabilities. But sediments are not continu-ously present where the folds are observed in the central Indian Basin, especially south of 1³S. This work demonstrates that sediments are not necessary to trigger lithospheric buck-ling at stresses below 300 MPa.
4. According to geophysical measurements [35], the large wavelength folds initiated about 8 Ma ago south of 1³S (scenario B), and de-veloped 4 Ma later north of 1³S (scenario A). Because the lithosphere is about 20 Ma older to the north, its competent layer is expected to be stronger and thicker: large-scale buckling should develop later, with a longer wavelength, but faster. The present models suggest that these di¡erent dates could be associated to a variation in space of the maximum deviatoric stresses. Seismic stratigraphy and pole rotation reconstructions show that the rate at which shortening occurs has not been constant since 20 Ma [29] and may even be periodic in time [35]. The value used here is O1= 2U10 16 s 1. But model 5c, in which the value is half as big, demonstrates that buckling is still possible within a reasonable time-scale. Varying the boundary conditions with time and the yield stress with space, constitute a next step in which such a complex situation could be handled.
5. Vertical deformation is partially accommo-dated by homogeneous thickening in the mod-el. Within 10 Ma, the BDT is predicted to deepen by 5 km, which would increase the thickness of the competent lithosphere by 12.5%. A number of other mechanisms not considered here may inhibit the increase in depth of the BDT, such as chemical phase transitions, £uid transport, di¡usion creep or Peierls power-law creep [11 14]. Continuing deformation will eventually isolate a single shear zone, thus initiating subduction. This process is strongly dependent on the rate of rheological softening (such as the decrease of the friction angle and/or cohesion with plastic deformation) [10]. Although it is well known that softening can have major consequences on the geometry of the deforming structure [7,10,50], insu¤cient data are presently
avail-able to allow formulation of a predictive mod-el.
6. The numerical models of this paper show the importance of the contrast in strength, which is equivalent to that of e¡ective viscosities, within the range 1036d/W _O16 104. The activated wavelength of buckling is approximately ¢ve times the thickness of the competent layer. The lateral di¡erential force is approximately 4 10U1012 N m 1. Those values are all con-sistent with previous estimates [2 5,18 20]. However, one must be aware that buckling in-stabilities produced in these simulations are in£u-enced by the ability of the code to handle
numer-ical instabilities. The more pre-existing
perturbations are present, the faster instabilities can grow. Therefore the results must be inter-preted with caution. Model 6 with a variable ther-mal conductivity demonstrates how sther-mall changes in the geotherm a¡ect the rheology of the rocks and the growth rate of buckling.
Acknowledgements
This work began at the University of Montpel-lier, was developed at the Institute of Geological and Nuceal Sciences in Wellington (contribution 1901) and was ¢nalized at the Minnesota Super-computer Institute with the IBM-SP system. It was supported by the New Zealand Foundation for Research, Science and Technology and by a grant from the Department of Energy to D. Yuen. It results from discussions with J. Marti-nod, N. Chamot-Rooke, G. Burov and Bobby Poliakov. I am immensely grateful to D. Yuen, L. Fleitout, K. Regenauer-Lieb and K.S. Krishna for comments on the manuscript. Due to the lim-ited number of references, I apologize for not having credited all useful contributions.[AC] References
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